1. Field of the Invention
This disclosure relates generally to numerical models for computer simulation of flow in a porous medium. More particularly, a method for simulating flow when a low viscosity fluid is injected into a formation containing more viscous resident fluid is provided, when viscous fingering and channeling of the injected fluid becomes important.
2. Background
This section is intended to introduce the reader to various aspects of the art, which may be associated with exemplary embodiments of the present invention, which are described and/or claimed below. This discussion is believed to be helpful in providing the reader with information to facilitate a better understanding of particular aspects of the present disclosure. Accordingly, it should be understood that these statements are to be read in this light, and not necessarily as admissions of prior art.
In the primary recovery of oil from a subterranean, oil-bearing formation or reservoir, it is usually possible to recover only a limited proportion of the original oil present in the reservoir. For this reason, a variety of supplemental recovery techniques are used to improve the displacement or recovery of oil from the reservoir rock. These techniques can be generally classified as thermal recovery methods (such as steam injection operations) and a-thermal recovery methods. A-thermal methods can involve injecting either immiscible fluids (e.g., water floods) or miscible fluids (e.g., CO2 floods or liquid solvent floods).
In miscible recovery operations, a soluble fluid (e.g. solvent) is injected into the reservoir to at least partially dissolve into the oil in place so that the oil can then be removed as a more highly mobile phase from the reservoir. The solvent is typically a light hydrocarbon such as liquefied petroleum gas (LPG), a hydrocarbon gas containing relatively high concentrations of aliphatic hydrocarbons in the C2 to C6 range, or carbon dioxide. Miscible recovery operations are normally carried out by a displacement procedure in which the solvent is injected into the reservoir through an injection well to displace the oil from the reservoir towards a production well from which the oil is produced. This provides effective displacement of the oil in the areas through which the solvent flows. Additionally, miscible recovery operations are sometimes cyclic in nature, where solvent is injected into the reservoir to invade and mix with the resident oil, and the resulting mixture is subsequently produced through the same well in which solvent was injected. Cyclic recovery operations typically consist of a number of injection and production cycles. For such miscible recovery operations, injected solvent often flows unevenly through the reservoir.
Because the solvent injected into the reservoir is typically substantially less viscous than the resident oil, the flow is often inherently unstable, which causes the solvent to finger and channel through the reservoir, leaving parts of the reservoir unswept. The unevenness of the sweep may be quite severe when the oil is highly viscous, such as the case of heavy oils and bitumens. Added to this fingering is the inherent tendency of a highly mobile solvent to flow preferentially through the more permeable rock sections.
The solvent's miscibility with the reservoir oil also affects its displacement efficiency within the reservoir. Some solvents, such as LPG, mix directly with reservoir oil in all proportions and the resulting mixtures remain single phase. Such solvent is said to be miscible on first contact or “first-contact miscible.” Other solvents used for miscible flooding, such as carbon dioxide or hydrocarbon gas, form two phases when mixed directly with reservoir oil. Therefore, they are not first-contact miscible. However, at sufficiently high pressure, in-situ mass transfer of components between reservoir oil and solvent forms a phase with a transition zone of fluid compositions that ranges from oil to solvent composition, and all compositions within the transition zone of this phase are contiguously miscible. Miscibility achieved by in-situ mass transfer of the components resulting from repeated contact of oil and solvent during the flow is called “multiple-contact” or dynamic miscibility. The pressure required to achieve multiple-contact miscibility is called the “minimum-miscibility pressure.” Solvents just below the minimum miscibility pressure, called “near-miscible” solvents, may recover oil nearly as well as miscible solvents.
Predicting performance of miscible recovery operations requires a realistic model representative of the reservoir. Numerical simulation of reservoir models is widely used by the petroleum industry as a method of using a computer to predict the effects of miscible displacement phenomena. In most cases, there is desire to model the transport processes occurring in the reservoir. What is being transported is typically mass, energy, momentum, or some combination thereof. By using numerical simulation, it is possible to reproduce and observe a physical phenomenon and to determine design parameters without actual laboratory experiments and field tests.
Reservoir simulation approximates the behavior of a real hydrocarbon-bearing reservoir from the performance of a numerical model of that reservoir. The objective of such simulations is to understand the complex chemical, physical, and fluid flow processes occurring in the reservoir sufficiently well to predict future behavior of the reservoir to maximize hydrocarbon recovery. Reservoir simulation often refers to the hydrodynamics of flow within a reservoir, but in a larger sense reservoir simulation can also refer to the total petroleum system which includes the reservoir, injection wells, production wells, surface flowlines, and surface processing facilities.
The principle of numerical simulation is to numerically solve equations describing a physical phenomenon by a computer, such as fluid flow. Such equations are generally ordinary differential equations and partial differential equations. These equations are typically solved by linearizing the equations and using numerical methods such as the finite element method, the finite difference method, the finite volume method, and the like. In each of these methods, the physical system to be modeled is divided into smaller gridcells or blocks (a set of which is called a grid or mesh), and the state variables continuously changing in each gridcell are represented by sets of values for each gridcell. In the finite difference method, an original differential equation is replaced by a set of algebraic equations to express the fundamental principles of conservation of mass, energy, and/or momentum within each gridcell and transfer of mass, energy, and/or momentum transfer between gridcells. These equations can number in the millions. Such replacement of continuously changing values by a finite number of values for each gridcell is called “discretization.” In order to analyze a phenomenon changing in time, it is necessary to calculate physical quantities at discrete intervals of time called timesteps, irrespective of the continuously changing conditions as a function of time. Time-dependent modeling of the transport processes proceeds in a sequence of timesteps.
In a typical simulation of a reservoir, solution of the primary unknowns, typically pressure, phase saturations, and compositions, are sought at specific points in the domain of interest. Such points are called “gridnodes” or more commonly “nodes.” Gridcells are constructed around such nodes, and a grid is defined as a group of such gridcells. The properties such as porosity and permeability are assumed to be constant inside a gridcell. Other variables such as pressure and phase saturations are specified at the nodes. A link between two nodes is called a “connection.” Fluid flow between two nodes is typically modeled as flow along the connection between them.
Compositional modeling of hydrocarbon-bearing reservoirs is necessary for predicting processes involving first-contact miscible, multiple-contact miscible, and near-miscible solvent injection. The oil and gas phases are represented by multicomponent mixtures. In such modeling, reservoir heterogeneity and viscous fingering and channeling cause variations in phase saturations and compositions to occur on scales as small as a few centimeters or less. A sufficiently fine-scale model can represent the details of these adverse-mobility injection behaviors. However, use of fine-scale models to simulate these variations is generally not practical because their fine level of detail places prohibitive demands on computational resources. Therefore, a coarse-scale model having far fewer gridcells is typically developed for reservoir simulation. Considerable research has been directed to developing models suitable for use in predicting performance of miscible recovery operations.
Numerical simulation of fluid flow in permeable hydrocarbon-bearing formations is a critical tool for optimizing the recoveries and economics of oil and gas production from subsurface reservoirs. Although several classes of numerical simulation schemes exist for such problems, finite-difference methods are by far the most commonly utilized in the petroleum industry. In such methods, the region to be simulated is populated with nodes at specific coordinates and sets of flow connections between neighboring nodes. Together the nodes and connections constitute a “simulation grid.”
Associated with each node is a subvolume of the simulation region (a “cell”) for which the conditions (e.g., pressure, composition, temperature, etc.) at the node apply. Various methods exist for assigning subvolumes to nodes. One common method is the Perpendicular Bisection (PEBI) method (see, e.g., C. L. Palagi and K. Aziz, “Use of Voronoi Grid in Reservoir Simulation,” SPE Advanced Technology Series, 2(2), 69-77, 1994). In this method, the line segments (for two-dimensional systems) or planar faces (for three-dimensional systems) defining the boundaries of a given subsection are orthogonal to and bisect the flow connections between neighboring nodes. The set of subvolumes forms a “cell pattern.” It is noted that any border associated with the connection between two nodes is actually a bounding face consisting of a finite surface area. For simplicity, these bounding faces are referred to here as boundary line segments in the two dimensional sense.
For many finite difference schemes the specific solution can be very dependent on the grid geometry, especially for coarse grids. Two types of errors can result. First, calculated solutions may change depending on the orientation of the grid (i.e., whether the fluid flow is aligned with the node connections or not). This is called a “grid orientation” effect. For example, fluid flow may be artificially forced along directions aligned with node connections. And second, calculated solutions may exhibit suppression of flow fingering or channeling due to artificially enhanced dispersion lateral to the flow direction. This phenomenon is termed “artificial dispersion” and in certain circumstances may be related to grid orientation. See FIGS. 1A-1D for an example of grid orientation and finger suppression due to lateral flow dispersion. FIGS. 1A-1D depict simulated concentration profiles at a specific time for a low-viscosity miscible fluid displacing (from the left) a fluid with a viscosity 150 times greater. In each case the grid is composed of rectangular cells but the orientations are rotated relative to the flow direction at angles of 0°, 15°, 30°, and 45°. Finger suppression due to lateral flow dispersion increases as the flow becomes further misaligned with the grid direction.
Overviews of grid orientation effects and solutions involving modifications to the calculation methods are given by Brand et al. (SPE 21228, “The Grid Orientation Effect in Reservoir Simulation”, 1991), Settari and Karcher (Journal of Canadian Petroleum Technology, “Simulation of enhanced recovery projects—the problems and pitfalls of current solutions”, 22-28, November-December 1985), and Haajizadeh et al., (SPE 62995, “Effects of Phase Behavior, Dispersion and Gridding on Sweep Patterns for Nearly Miscible Gas Displacement”, 2000).
Ideally, a simulation grid should not artificially favor flow in certain directions nor artificially enhance flow dispersion. Certain methods that address one of these problems can aggravate the other. For example, a cell pattern consisting of regular hexagons (hex-grids) may significantly reduce grid orientation effects as compared with square cell patterns. Hex-grids, however, have no principal flow directions (as will be discussed below) and thus tend to enhance lateral flow dispersion. This is typically not a major concern for systems where a less mobile fluid displaces a more mobile fluid (e.g., waterflooding a reservoir containing light oil). However, enhanced lateral dispersion may lead to serious prediction errors for systems where a more mobile fluid displaces or invades a less mobile fluid (e.g., solvent extraction of bitumen or CO2 flooding of an oil reservoir). In such systems, fingering behavior is prone to occur where the more mobile fluid attempts to displace or invade the less mobile fluid. Enhanced lateral dispersion may artificially “blur-out” such fingers in a simulation, as illustrated in FIGS. 2A-2B, and thus fingers may inappropriately grow slower or not at all. FIGS. 2A-2B depicts simulated concentration profiles at a specific time for a low viscosity miscible fluid displacing (from the left) a fluid with a viscosity 150-times greater. The enhanced lateral dispersion in the hex-grids of FIG. 2A leads to inappropriately wide and short fingers compared to the rectangular grid of FIG. 2B. It is noted that the flow is aligned with the grid direction for the rectangular grid case.
A number of methods have been proposed which address both artificial grid orientation effects and flow dispersion effects (e.g., multipoint flux approximation methods and mixed finite element methods) (J. E. Aarnes, T. Gimse, and K.-A. Lie, “An Introduction to the Numerics of Flow in Porous Media using Matlab” in Geometric Modeling, Numerical Simulation and Optimization: Applied Mathematics at SINTEF, G. Hasle et al. (eds.), 2007), but typically they reduce computation speed or are not available in commonly used simulators. The methods described in the current invention can be readily implemented in simulators utilizing the common two-point flux approximation method.
A primary cause of grid orientation effects and enhanced lateral artificial dispersion can be physically understood in terms of fluid flow not being aligned with a “principal flow direction” of the chosen finite difference grid. “Principal flow direction” can also be designated as “low-dispersive flow direction,” which describes a flow direction in the grid in which artificial dispersion is minimized. “Principal flow directions,” or “low-dispersive flow directions” are those flow directions where: (1) the flow has no velocity component causing a portion of it to cross cell boundaries defining a channel and (2) the flow follows a channel of substantial constant width defined by cell boundaries combining to form straight lines across the grid—as illustrated by FIGS. 3A and 3B. Note that for certain grid patterns two or more cells may span a principal flow direction channel and thus dispersion may be enhanced within the channel but not across the boundaries of the channel. In radial grids, the channels are between radial lines. In other grids, the channels may be between parallel lines.
If a fluid flow direction is not aligned with a principal flow direction of the finite difference grid, the flow will repeatedly split and laterally disperse as it moves through a progression of cells. For example, a standard rectangular grid, as shown in FIG. 3A, has two principal flow directions (0° and 90°) and a standard triangular grid has three principal flow directions (0°, 60°, 120°), as shown in FIG. 3B. The degree to which the lateral dispersion occurs is proportional to the angular offset from the closest principal flow direction. Thus, grid orientation effects and enhanced dispersion effects are related. Additionally, artificially high lateral dispersion can occur if the flow directions correspond to cell channels which become progressively wider, which is typically the case in radial grids, as illustrated in FIG. 4. Thus, radial grids of the type illustrated in FIG. 4 are considered not to have any principal flow directions, as defined herein. Embodiments of the invention describe modified radial grids with cells forming channels of substantially constant width, or at least cells that have a limited variation of width.
Lateral dispersion may suppress flow fingering or channeling behavior, and thus if the dispersion is artificially enhanced due to the choice of cell pattern, physically incorrect behavior may be predicted for certain systems. Viscous fingering may occur in systems with a mobility ratio greater than one (i.e., displacing fluid mobility greater than displaced fluid mobility), where the mobility of a fluid is defined as the ratio of permeability it experiences to its viscosity. Such adverse mobility ratios can occur in commercially important light and heavy oil systems such as CO2 flooding, water flooding, low concentration polymer flooding, and solvent injection heavy oil recovery methods. In such systems the mobility ratio may be greater than 1, 10, or even 100. Non-viscous types of fingering, or channeling, can occur in certain systems including those in systems which undergo fracturing (e.g., cyclic steam stimulation) and in systems with certain geological heterogeneities.
Chong, et al. (SPE88617, “A Unique Grid-Block System for Improved Grid Orientation”, 2004), discuss a novel gridding method for reduction of grid orientation artifacts utilizing square and octagon cells. The method however is not suitable for viscous fingering systems since the grid does not address artificial lateral dispersion. Huh, et al. (U.S. Pat. No. 7,006,959) propose an approximation of mathematically splitting each cell into two cells to capture the fact that viscous fingers only displace fluid from a portion of the volume. The approximation, however requires additional simulation coding and involves adjustable parameters requiring tuning. Watts and Silliman (C. C. Mattax (ed.) and R. L. Dalton (ed.), Reservoir Simulation, SPE Monograph Series, Chapter 5, 1990) proposed a gridding method that results in four principal flow directions. The method is applicable in common simulators. The method, however, is not optimal.
Hence, a better method for modeling flow in a porous media is needed.